Finite Group Databasebeta

← #31791044

Subgroups — #31791044

Order 17536.

384 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22578020C211yesmaximal, characteristic, complemented
2#22577886C211yesmaximal, characteristic, complemented
3#22577739C211yesmaximal, characteristic, complemented
4#22577714C211yesmaximal, characteristic, complemented
5#22577696C211yesmaximal, characteristic, complemented
6#22577681C211yesmaximal, characteristic, complemented
7#22577623C211yesmaximal, characteristic, complemented
8#22577616C211yesmaximal, characteristic, complemented
9#22577565C211yesmaximal, characteristic, complemented
10#22577504C211yesmaximal, characteristic, complemented
11#22577500C211yesmaximal, characteristic, complemented
12#22577495C211yesmaximal, characteristic, complemented
13D8×Dic137C211yesmaximal, characteristic, complemented
14#22577461C211yesmaximal, characteristic, complemented
15#449×C137C211yesmaximal, characteristic
16C2×#14555500C2211yescharacteristic
17C2×#14555498C2211yescharacteristic
18#17984087C2211yescharacteristic
19#17984081—21no—
20#17984075—21no—
21#17984075C2211yescharacteristic
22#17984069—21no—
23#17984067C2211yescharacteristic
24#17984065—21no—
25Q8×Dic137C2211yescharacteristic
26Q8×Dic137C2211yescharacteristic
27#17984053C2211yescharacteristic
28D4×Dic137—21no—
29D4×Dic137—21no—
30D4×Dic137C2211yescharacteristic
31D4×Dic137C2211yescharacteristic
32D4×Dic137C2211yescharacteristic
33#17984047—21no—
34#17984047C2211yescharacteristic
35C2×C4⋊Dic137—21no—
36C2×C4⋊Dic137C2211yescharacteristic
37#17984022C2211yescharacteristic
38#17984022C2211yescharacteristic
39#17984021C2211yescharacteristic
40#17984021C2211yescharacteristic
41#17984017C2211yescharacteristic
42#17984017C2211yescharacteristic
43#17984013C2211yescharacteristic
44#17984013C2211yescharacteristic
45#17984012C2211yescharacteristic
46#17984009C2211yescharacteristic
47#17984009C2211yescharacteristic
48#17984009C2211yescharacteristic
49C22⋊#11764040C2211yescharacteristic
50#17983998C2211yescharacteristic
51C8×Dic137C2211yescharacteristic
52C2×#41×C137C2211yescharacteristic
53C2×SD16×C137C2211yescharacteristic
54C2×D8×C137C2211yescharacteristic
55#121×C137C2211yescharacteristic
56#103×C137C2211yescharacteristic
57#102×C137C2211yescharacteristic
58C22⋊C8×C137C2211yescharacteristic
59C2×Dic274—21no—
60C2×Dic274C2311yescharacteristic
61C22×Dic137—21no—
62C22×Dic137—21no—
63C22×Dic137—21no—
64C22×Dic137—21no—
65#14555500—22no—
66#14555498—22no—
67C2×#11764040C2311yescharacteristic
68C2×#11764040C2311yescharacteristic
69C4⋊Dic137—21no—
70C4⋊Dic137—21no—
71C4⋊Dic137—21no—
72C4⋊Dic137D411yescharacteristic
73C4⋊Dic137D411yescharacteristic
74C4⋊Dic137C2311yescharacteristic
75C4⋊Dic137C2311yescharacteristic
76C4⋊Dic137C2311yescharacteristic
77#14555489—21no—
78#14555489—21no—
79#14555489—21no—
80C22⋊Dic137—21no—
81C22⋊Dic137—21no—
82C22⋊Dic137—21no—
83C22⋊Dic137—21no—
84C22⋊Dic137—21no—
85C22⋊Dic137—21no—
86C22⋊Dic137D411yescharacteristic
87C22⋊Dic137D411yescharacteristic
88C4×Dic137—21no—
89C4×Dic137—21no—
90C4×Dic137—21no—
91C4×Dic137C2311yescharacteristic
92C4×Dic137C2311yescharacteristic
93#41×C137—22no—
94#41×C137—22no—
95C2×Q8×C137C2311yescharacteristic
96C2×D4×C137—21no—
97C2×D4×C137—21no—
98C2×D4×C137C2311yescharacteristic
99C2×D4×C137C2311yescharacteristic
100C22×C4×C137—21no—
101C22×C4×C137C2311yescharacteristic
102SD16×C137—22no—
103D8×C137—22no—
104C2×C8×C137C2311yescharacteristic
105C2×C8×C137C2311yescharacteristic
106C4⋊C4×C137C2311yescharacteristic
107C22⋊C4×C137—21no—
108Dic274—21no—
109Dic274—21no—
110Dic274—81no—
111C2×Dic137—21no—
112C2×Dic137—21no—
113C2×Dic137—21no—
114C2×Dic137—21no—
115C2×Dic137—81no—
116C2×Dic137—81no—
117C2×Dic137—81no—
118C2×Dic137—81no—
119C2×Dic137C2×D411yescharacteristic
120C2×Dic137C2×D411yescharacteristic
121C2×Dic137C2×D411yescharacteristic
122C2×Dic137C2×D411yescharacteristic
123#11764040—21no—
124#11764040—21no—
125C23×C137—21no—
126C23×C137—21no—
127C23×C137C2×D411yescharacteristic
128Q8×C137—21no—
129Q8×C137#4112yes—
130D4×C137—24no—
131D4×C137—21no—
132D4×C137—21no—
133D4×C137—21no—
134D4×C137—81no—
135D4×C137#4112yes—
136C2×C4×C137—24no—
137C2×C4×C137—21no—
138C2×C4×C137—21no—
139C2×C4×C137—21no—
140C2×C4×C137—21no—
141C2×C4×C137C2×D411yescharacteristic
142C2×C4×C137C2411yescharacteristic
143C8×C137—21no—
144C8×C137—21no—
145C22×C137—21no—
146C22×C137—22no—
147C22×C137—21no—
148C22×C137—21no—
149C22×C137—41no—
150C22×C137—41no—
151C22×C137—41no—
152C22×C137—41no—
153C22×C137—41no—
154C22×C137C22×D411yescharacteristic
155C4×C137—21no—
156C4×C137—22no—
157C4×C137—41no—
158C4×C137C2×#4111yescharacteristic
159C4×C137C2×#4111yescharacteristic
160Dic137—21no—
161Dic137—21no—
162Dic137—21no—
163Dic137—21no—
164Dic137—41no—
165Dic137—41no—
166Dic137—41no—
167Dic137—41no—
168C2×C137—41no—
169C2×C137—42no—
170C2×C137—81no—
171C2×C137#52511yescharacteristic
172C2×C137#57611yescharacteristic
173C2×C137#57811yescharacteristic
174C137#312411yescharacteristic, complemented, Sylow
175#3124—1371nomaximal, Sylow
176#542—1371no—
177#520—1371no—
178#484—1371no—
179#480—1371no—
180#477—1371no—
181#474—1371no—
182#464—1371no—
183#463—1371no—
184#452—1371no—
185#449D13711yescharacteristic
186#440—1371no—
187#440—1371no—
188C4×Q16—1371no—
189C4×D8—1371no—
190#433—1371no—
191C2×#41C2×D13711yescharacteristic
192C2×Q16—1371no—
193C2×SD16—1371no—
194C2×SD16C2×D13711yescharacteristic
195C2×D8C2×D13711yescharacteristic
196C4⋊Q8—1371no—
197#126—2741no—
198#124—2741no—
199#124—1371no—
200#123—2741no—
201C22⋊Q8—2741no—
202C22⋊Q8—1371no—
203#121C2×D13711yescharacteristic
204C4×Q8—1371no—
205C4×Q8—1371no—
206C4×Q8—1371no—
207C4×D4—2741no—
208C4×D4—2741no—
209C4×D4—1371no—
210C4×D4—1371no—
211C4×D4—1371no—
212#117—2741no—
213#117—1371no—
214C2×C4⋊C4—2741no—
215C2×C4⋊C4—1371no—
216#107—1371no—
217#107—1371no—
218#107—1371no—
219#107—1371no—
220C4⋊C8—1371no—
221C4⋊C8—1371no—
222#103—1371no—
223#103—1371no—
224#103—1371no—
225#103C2×D13711yescharacteristic
226#102—1371no—
227#102—1371no—
228#102—1371no—
229#102C2×D13711yescharacteristic
230C22⋊C8—1371no—
231C22⋊C8C2×D13711yescharacteristic
232#97—1371no—
233C4×C8—1371no—
234#41—22no—
235#41—22no—
236C2×Q8—2741no—
237C2×Q8—1371no—
238C2×Q8C22×D13711yescharacteristic
239C2×D4—21no—
240C2×D4—21no—
241C2×D4C22×D13711yescharacteristic
242C2×D4C22×D13711yescharacteristic
243C22×C4—21no—
244C22×C4—2741no—
245C22×C4—2741no—
246C22×C4—2741no—
247C22×C4—2741no—
248C22×C4C22×D13711yescharacteristic
249Q16—2742no—
250SD16—22no—
251SD16—2742no—
252D8—22no—
253C2×C8—1371no—
254C2×C8—1371no—
255C2×C8C22×D13711yescharacteristic
256C2×C8C22×D13711yescharacteristic
257C4⋊C4—2741no—
258C4⋊C4—2741no—
259C4⋊C4—2741no—
260C4⋊C4—2741no—
261C4⋊C4—2741no—
262C4⋊C4—2741no—
263C4⋊C4—1371no—
264C4⋊C4—1371no—
265C4⋊C4—1371no—
266C4⋊C4—1371no—
267C4⋊C4—1371no—
268C4⋊C4C22×D13711yescharacteristic
269C22⋊C4—21no—
270C22⋊C4—2741no—
271C22⋊C4—2741no—
272C22⋊C4—2741no—
273C22⋊C4—2741no—
274C22⋊C4—2741no—
275C22⋊C4—2741no—
276C22⋊C4—1371no—
277C22⋊C4—1371no—
278C42—2741no—
279C42—2741no—
280C42—2741no—
281C42—1371no—
282C42—1371no—
283C23—21no—
284C23—21no—
285C23D4×D13711yescharacteristic
286Q8—21no—
287Q8—2741no—
288Q8—2741no—
289Q8—10961no—
290Q8#1455550912yes—
291D4—24no—
292D4—21no—
293D4—21no—
294D4—21no—
295D4—81no—
296D4#1455550912yes—
297C2×C4—24no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—2741no—
303C2×C4—2741no—
304C2×C4—2741no—
305C2×C4—2741no—
306C2×C4—10961no—
307C2×C4—10961no—
308C2×C4—10961no—
309C2×C4—10961no—
310C2×C4—1371no—
311C2×C4—1371no—
312C2×C4—1371no—
313C2×C4—1371no—
314C2×C4D4×D13711yescharacteristic
315C2×C4C23×D13711yescharacteristic
316C8—21no—
317C8—21no—
318C8—2741no—
319C8—2741no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—22no—
324C22—41no—
325C22—41no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22C2×D4×D13711yescharacteristic
330C4—21no—
331C4—22no—
332C4—41no—
333C4—2741no—
334C4—2741no—
335C4—2741no—
336C4—2741no—
337C4—5481no—
338C4—5481no—
339C4—5481no—
340C4—5481no—
341C4C2×#1455550911yescharacteristic
342C4C2×#1455550911yescharacteristic
343C2—41no—
344C2—42no—
345C2—81no—
346C2#2257790911yescharacteristic
347C2#2257820711yescharacteristic
348C2#2257821711yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting, prime radical#449×C137C2
socleC22×C137C22×D4
hypercenter#449D137
hypocenterC137#3124
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #22578020C2semidirect
2. #22577886C2semidirect
3. #22577739C2semidirect
4. #22577714C2semidirect
5. #22577696C2semidirect
6. #22577681C2semidirect
7. #22577623C2semidirect
8. #22577616C2semidirect
9. #22577565C2semidirect
10. #22577504C2semidirect
11. #22577500C2semidirect
12. #22577495C2semidirect
13. D8×Dic137C2semidirect
14. #22577461C2semidirect
174. C137#3124semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#225780202
#225778862
#225777392
#225777142
#225776962
#225776812
#225776232
#225776162
#225775652
#225775042
#225775002
#225774952
D8×Dic1372
#225774612
#449×C1372
#3124137